The Secure Metric Dimension of the Globe Graph and the Flag Graph

Abstract

Let G = (V, E) be a connected, basic, and finite graph. A subset $T=\left\{u_1,u_2,\dots ,u_k\right\}$ of V(G) is said to be a resolving set if for any y ∈ V(G), the code of y with regards to T, represented by $C_T\left(y\right)$, which is defined as $C_T\left(y\right)=d\left(u_1,y\right),d\left(u_2,y\right),\dots ,d\left(u_k,y\right)$, is different for various y. The dimension of G is the smallest cardinality of a resolving set and is denoted by dim(G). If, for any t  ∈  V – S, there exists r ∈ S such that $\left(S–\left\{r\right\}\right)\cup \left\{t\right\}$ is a resolving set, then the resolving set S is secure. The secure metric dimension of 𝐺 is the cardinal number of the minimum secure resolving set. Determining the secure metric dimension of any given graph is an NP-complete problem. In addition, there are several uses for the metric dimension in a variety of fields, including image processing, pattern recognition, network discovery and verification, geographic routing protocols, and combinatorial optimization. In this paper, we determine the secure metric dimension of special graphs such as a globe graph $G{l}_n$, flag graph $F{l}_n$, H- graph of path $P_n$, a bistar graph $B_{n,n}^2$, and tadpole graph $T_{3,m}$. Finally, we derive the explicit formulas for the secure metric dimension of tadpole graph $T_{n,m}$, subdivision of tadpole graph $S\left(T_{3,m}\right)$, and subdivision of tadpole graph $S\left(T_{n,m}\right)$.